R\'enyi divergences and binary state discrimination error exponents for fermionic quasi-free states

Abstract:The trade-off relations between the two types of error probabilities in binary i.i.d. quantum state discrimination can be expressed by single-copy formulas in terms of the Petz-type and the sandwiched Rényi divergences of the two states representing the two hypotheses. In the non-i.i.d. setting, the error exponents can usually be expressed in terms of regularized Rényi divergences, which do not admit explicit formulas in general. Here, we consider a class of states, translation-invariant and gauge-invariant quasifree states on doubly infinite fermionic chains, and give explicit formulas for a wide range of regularized Rényi divergences between such states, including $(\alpha,z)$, log-Euclidean, geometric, measured, and the recently introduced integral Rényi divergences. We show that the case where there is a single mode at each lattice site becomes asymptotically classical, with all the different types of regularized Rényi divergences being equal, while in the case of multiple modes per site, non-commutativity persists under regularization, and for any fixed $\alpha$, the regularized Rényi $(\alpha,z)$-divergences give different regularized values for different $z$ parameters in general. We also generalize a previous construction from [Bunth, Maróti, Mosonyi, Zimborás, Lett.~Math.~Phys.~113:(7), 2023] to the case of multiple modes per lattice site to obtain a large class of states exhibiting super-exponential decay of the discrimination error probabilities.

40 pages. v2: Slightly improved presentation, several typos corrected, main results unchanged

Quantum Physics (quant-ph); Information Theory (cs.IT); Mathematical Physics (math-ph)

arXiv:2605.31379 [quant-ph]

(or arXiv:2605.31379v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2605.31379 Focus to learn more arXiv-issued DOI via DataCite

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